Canonical Correlation: EquationsPsy 524Andrew AinsworthData for Canonical Correlations{ CanCorr actually takes raw data and computes a correlation matrix and uses this as input data.{ You can actually put in the correlation matrix as data (e.g. to check someone else’s results)Data{ The input correlation set up is:xx xyyx yyRRRREquations{ To find the canonical correlations:z First create a canonical input matrix. To get this the following equation is applied:11yy yx xx xyRRRRR−−=Equationsz To get the canonical correlations, you calculate the eigenvalues of R and take the square rootci irλ=Equationsz In this context the eigenvaluesrepresent percent of overlapping variance accounted for in all of the variables by the two canonical variates{ i.e. it is the squared correlationEquations{ Testing Canonical Correlationsz Since there will be as many CanCorrsas there are variables in the smaller set, not all will be meaningful (or useful).Equations{ Wilk’s Chi Square test – tests whether a CanCorr is significantly different than zero.2xy111ln2Where N is number of cases, k is number of x variables andk is number of y variables(1 )Lamda, Λ, is the product of difference between eigenvalues and 1, gexymmmiikkNχλ=++=− − − ΛΛ= −∏nerated across m canonical correlations.Equations{ From the text example - For the first canonical correlation:222(1 .84)(1 .58) .0722181 ln.072(4.5)( 2.7) 12.15()()(2)(2)4xydf k kχχΛ= − − =++=− − −=− − ====Equations{ The second CanCorr is tested as122(1 .58) .4222181 ln.422(4.5)( .87) 3.92( 1)( 1) (2 1)(2 1) 1xydf k kχχΛ= − =++=− − −=− − ==− −=− −=Equations{ Canonical Coefficients z Two sets of Canonical Coefficients are required{ One set to combine the Xs{ One to combine the Ys{ Similar to regression coefficientsEquations1/ 21/ 2yˆ()'Where ( ) ' the transpose of the inverse of the "special" matrix ˆform of square root that keeps all of the eigenvalues positive and is a normalized matrix of eigen vectors fyyyyyyBR BRisB−−=-1 *x*or yyBWhere is from above dividing each entry by their corresponding canonical correlation.xx xy yyyRRBBB=Equations{ Canonical Variate Scoresz Like factor scores (we’ll get there later)z What a subject would score if you could measure them directly on the canonical variatexxyyXZBYZB==Equations{ Matrices of Correlations between variables and canonical variates; also called loadings or loading matricesxxxxyyyyARBARB==Equations Canonical Variate Pairs First Second TS -.74 .68 First Set TC .79 .62 BS -.44 .90 Second SetBC .88 .48Equations{ Percent of variance in a single variable accounted for by it’s own canonical variatez This is simply the squared loading for any variablez e.g. The percent of variance in Top Shimmies explained by the first canonical variate is -.742≈ 55%Equations{ Redundancyz Within –Averagepercent of variance in a set of variables explained by their own canonical variate1212122( .74) (.79).582xykixcxcixkiycyciyxcapvkapvkpv====−+==∑∑Equations{ Redundancyz Across – average percent of variance in the set of Xs explained by the Y canonical variate and vice versa1222()()( .74) .79(.84) .482cxyrd pv
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